hydraulic criticality of the exchange flow through …...hydraulic criticality of the exchange flow...

Hydraulic Criticality of the Exchange Flow through the Strait of Gibraltar

GIANMARIA SANNINO

Ocean Modelling Unit, ENEA, Rome, Italy

LAWRENCE PRATT

Woods Hole Oceanographic Institution, Woods Hole, Massachusetts

ADRIANA CARILLO

Ocean Modelling Unit, ENEA, Rome, Italy

(Manuscript received 17 June 2008, in final form 9 January 2009)

ABSTRACT

The hydraulic state of the exchange circulation through the Strait of Gibraltar is defined using a recently

developed critical condition that accounts for cross-channel variations in layer thickness and velocity, applied

to the output of a high-resolution three-dimensional numerical model simulating the tidal exchange. The

numerical model uses a coastal-following curvilinear orthogonal grid, which includes, in addition to the Strait

of Gibraltar, the Gulf of Cadiz and the Alboran Sea. The model is forced at the open boundaries through the

specification of the surface tidal elevation that is characterized by the two principal semidiurnal and two

diurnal harmonics: M2, S2, O1, and K1. The simulation covers an entire tropical month.

The hydraulic analysis is carried out approximating the continuous vertical stratification first as a two-layer

system and then as a three-layer system. In the latter, the transition zone, generated by entrainment and

mixing between the Atlantic andMediterranean flows, is considered as an active layer in the hydraulic model.

As result of these vertical approximations, two different hydraulic states have been found; however, the

simulated behavior of the flow only supports the hydraulic state predicted by the three-layer case. Thus,

analyzing the results obtained by means of the three-layer hydraulic model, the authors have found that the

flow in the strait reaches maximal exchange about 76% of the tropical monthlong period.

1. Introduction

The Strait of Gibraltar is a narrow and shallow

channel 60 km long and 20 km wide, characterized by a

complex system of contractions and sills (Fig. 1). In the

eastern part of the strait a deep channel is present, called

Tarifa Narrows (TN), characterized by a mean cross

section of about 18 km and a depth of more than 800 m.

To the west the bottom abruptly rises, reaching the min-

imum depth of the whole strait (284 m) at Punta Ca-

marinal, determining the so-called Camarinal Sill (CS);

then the bathymetry is characterized by the presence of

a submarine ridge called Majuan Bank (MB in Fig. 1)

dividing the cross section into two channels. The northern

channel has a maximum depth of only 250 m, while the

southern channel has a maximum depth of 360 m that

is actually a relative minimum depth for the main along-

strait channel in the western part of the strait. This topo-

graphic point, called Espartel Sill (ES in Fig. 1), represents

the last topographic constriction for the Mediterranean

Outflow. Through this channel more than 80% of Medi-

terranean Water flows into the Gulf of Cadiz (Sanchez-

Roman et al. 2009).

The mean circulation within the Strait of Gibraltar is

described as an inverse estuarine circulation (Stommel

and Farmer 1953), characterized by a two-way exchange,

with an upper flow of fresh (SA ’ 36.2 psu) and warm

Atlantic water spreading in the Mediterranean basin

and a lower flow of cold and salty Mediterranean Water

(SM ’ 38.4 psu) sinking in the North Atlantic down to a

depth of ;1000 m where it becomes neutrally buoyant

(Baringer and Price 1997; Ambar et al. 2002). While the

excess evaporation over precipitation and river runoff

that takes place in the Mediterranean Sea drives this

Corresponding author address: Gianmaria Sannino, via Anguil-

larese 301, Ocean Modelling Unit, ENEA, 00030 Rome, Italy.

E-mail: [email protected]

NOVEMBER 2009 SANN INO ET AL . 2779

DOI: 10.1175/2009JPO4075.1

� 2009 American Meteorological Society

mean circulation, its magnitude and hydrological prop-

erties strongly depend on the physical configuration of

the strait (Bryden and Stommel 1984). In fact, it is well

known that the Strait of Gibraltar is a place where the

water exchange is subject to hydraulic control. However,

a key issue that is still an open question regards the

number and location of the hydraulic controls. These

have a crucial role in forcing the strait dynamics toward

one of the following two possible regimes: maximal and

submaximal. If the exchange is subject to one hydraulic

control in the western part of the strait, the regime is

called submaximal while, if the flow exchange is also

controlled in the eastern part of the strait along TN, the

regime is called maximal. The two regimes have differ-

ent implications for property fluxes, response time, and

other physical characteristics of the coupled circulation

in the strait and Mediterranean Sea. The maximal re-

gime can be expected to have larger heat, salt, and mass

fluxes and to respond more slowly to changes in strati-

fication and thermohaline forcing within the Mediter-

ranean Sea and the North Atlantic Ocean (Reid 1979).

Progress in understanding the hydraulic regime in the

Strait of Gibraltar was made byArmi and Farmer (1988)

and Farmer and Armi (1988, hereafter FA88) who an-

alyzed data collected during the Gibraltar Experiment

(Kinder and Bryden 1987). They approximated the

vertical structure of the exchange flow as a two-layer

system and showed the presence of four controls: two

permanent and two episodic. The first permanent con-

trol was located west of ES, while the second was sited

within TN, moving cyclically toward the east in accor-

dance with the eastward-traveling internal bore released

by CS. The two episodic controls were located over the

two sills Espartel and Camarinal. According to these

findings the exchange flow was continuously in a maxi-

mal regime during the period of observation (April

1986).

Many papers have subsequently dealt with the appli-

cability of the hydraulic theory to the exchange flow in

the Strait of Gibraltar, focusing their analysis on the

number and location of the hydraulic controls. Among

others, see Bormans et al. (1986), Garrett et al. (1990),

Bryden and Kinder (1991), Garcıa-Lafuente et al. (2000),

and Send and Baschek (2001) for the experimental

and analytical approach and Izquierdo et al. (2001) and

Sannino et al. (2002) for numerical modeling studies. All

of this work treats the exchange as a two-layer system.

However, as demonstrated by Bray et al. (1995, here-

inafter BR95), the two-way exchange is strongly af-

fected by entrainment and mixing between the Atlantic

and Mediterranean waters. Entrainment and mixing are

sufficiently strong (Wesson and Gregg 1994) to lead to

the formation of a thick interfacial layer where density

and velocity change gradually in the vertical. The pres-

ence of this thick layer complicates the estimation of the

hydraulic state by means of the two-layer hydraulic

theory; in fact, depending on the way the currents and

the thickness of both layers are defined, the values of the

calculated hydraulic state may vary significantly (Send

and Baschek 2001). Thus, in order to limit the arbitrar-

iness in defining the interface, Sannino et al. (2007)

proposed explicitly taking into account the three-layer

FIG. 1. Map of the Strait of Gibraltar showing the main topographic features—ES: Espartel

Sill, TB: Tangier Basin, CS: Camarinal Sill, and TN: Tarifa Narrows. MB indicates the sub-

marine ridge of Majuan Bank, located north of ES, dividing the cross section across ES in two

channels. Other geographic locations referred to in the text are also shown.

2780 JOURNAL OF PHYS ICAL OCEANOGRAPHY VOLUME 39

approximation introduced by BR95; in particular, they

computed the hydraulic state applying the three-layer

one-dimensional approximation derived by Smeed (2000).

Results showed the presence of a periodic control over CS

as for the two-layer case, but also a permanent super-

critical region close to the north shore of TN near the

Strait of Gibraltar.

It should be noted that all of the previous attempts to

study the hydraulic criticality of the Strait of Gibraltar

neglected the cross-strait variation of the layer thickness

and velocity. However, even when idealized as a two- or

three-layer system, the Gibraltar exchange flow is two-

dimensional since cross-strait variations in velocity and

layer thickness generally exist. The purpose of this paper

is to assess the criticality of the exchange flow for both

the two- and three-layer cases, including these cross-

strait variations. To this end, the recent hydraulic cri-

terion model developed by Pratt (2008) for an arbitrary

number of layers, allowing for cross-channel variations

in both thickness and velocity, has been adapted to the

Strait of Gibraltar. The new criterion determines the

hydraulic criticality of the flow as a whole across any

section, and thereby removes any ambiguity that arises

when the flow varies from being locally subcritical to

supercritical across a section. The criterion has been

used in synergy with a high-resolution three-dimensional

numerical model designed to provide a very detailed

description of the tidal exchange in the strait.

The present paper is organized as follows: the appli-

cation of the hydraulic criterion to the strait is presented

in section 2, while the numerical model used is described

in section 3. Results obtained for the two- and three-

layer system are shown in section 4, while conclusions

are discussed in section 5.

2. The hydraulic criterion

a. Two-layer system

The hydraulic state of a steady, Boussinesq, two-layer

flow with a rigid lid is determined by the value of the

composite Froude number (Armi 1986)

G2 5F 21 1F 2

2 , (1)

where Fn 5 un/g9Hn, un and Hn are the velocity and

thickness of layer n, and g9 is the reduced gravity. The

flow is considered subcritical, critical, or supercritical,

according to whether G2 , 1, G2 5 1, or G2 . 1, re-

spectively. A generalization to three-layer flow has been

obtained by Smeed (2000). In both cases the flow is as-

sumed to be one-dimensional so that cross-strait varia-

tions in layer thickness and velocity are not allowed.

In a two-layer system allowing cross-channel varia-

tions in layer velocities and thicknesses, the value of G2

will vary across the channel and its value at any point is

not a conclusive indicator of the hydraulic state (sub-

critical or supercritical) of the flow as a whole at that

section. Rather, the hydraulic state depends on the prop-

agation speed of long waves that exist across the entire

section and satisfy bottom and sidewall boundary con-

ditions. This complication can be dealt with when the

layer thicknesses, but not the velocity, are allowed to

vary across the strait. The procedure, described by

Baines (1995), has been used to investigate the three-

layer hydraulics of the Bab alMandab (Pratt et al. 1999).

When the layer depths and velocities vary, the criticality

of the flow can be determined by a criterion developed

by Pratt (2008). The procedure can be applied to an

arbitrary number of layers, and we will do so for two-

and three-layer representations of the flow through the

Strait of Gibraltar. There are some caveats in the in-

terpretation of the results, and these are described next.

For a two-layer system (Fig. 2a) the condition for

critical flow is given by

G2w 5

1

w�1I

ðw0

(g9H1/u21) dy

11

w�1I

ðyRyL

(g9H2/u22) dy

5 1,

(2)

where w is the width of the surface and wI the width of

the interface. The generalized composite Froude number

FIG. 2. Definition sketch for (a) two-layer and (b) three-layer flow.


defined in (2) reduces to the familiarG2 whenHn and unare constants and w 5 wI.

b. Three-layer system

For a three-layer system (Fig. 2b) the generalized

critical condition is

~F2

1 11� r

r1

w3

w2

� �~F2

2 1~F2

3 � w3

w2

~F2

1~F2

2 � ~F2

1~F2

3

� 1� r

r~F2

2~F2

3 5 1, (3)

where

~F2

1 51

w2

ðy1Ry1L

g921H

1

u21dy

1

!�1

,

~F2

2 51

w2

ðy2Ry2L

g932H

2

u22dy

2

!�1

,

~F2

3 51

w3

ðy3Ry3L

g932H

3

u23dy

3

!�1

; (4)

g921 5 g(r2 � r1)/r, g932 5 g(r3 � r2)/r, r 5 (r2 � r1)/

(r3 � r1), and wn is the width of the interface overlying

layer n. Note that ~F2

1~F2

2 and ~F2

3 can be interpreted as

generalized versions of layer Froude numbers. It can be

shown that (3) reduces to the condition derived by

Smeed (2000) for the three-layer case when Hn and unare each uniform and w1 5 w2 5 w3. It is tempting to

think of the lhs of (3) as a composite Froude number

and apply the same interpretations as for the two-layer

composite Froude number G2 (viz., that the flow is

subcritical, critical, or supercritical according toG2 , 1,

G25 1, orG2. 1). However, the left-hand side is clearly

negative when all three individual Froude numbers are

large and therefore a simple interpretation along the

lines of the two-layer version is less obvious.

In a single-layer system with depthH and velocity u it

is possible to decompose the phase speed into an ad-

vective part u and an intrinsic propagation partffiffiffiffiffiffiffigH

p,

equal to the propagation speed in a resting fluid. In a

multilayered system, this decomposition is possible only

if the flow lacks shear, both horizontally and vertically.

Thus, the layer Froude numbers in (4) are defined for

convenience and cannot generally be thought of as ratios

of intrinsic to advection speeds. To determine whether a

particular state is subcritical, supercritical, or critical, it

is helpful to rewrite (3) as

w3

w2

~F2

2 5� ( ~F2

1 � 1)( ~F2

3 � 1)

( ~F2

1 � 1)1b( ~F2

3 � 1), (5)

where

b5w

2(1� r)

w3r

.

Equation (5) defines a two-leafed surface in the space

~F2

1 ,w

3

w2

~F2

2 ,~F2

3

� �,

as shown in Fig. 3. The first leaf hovers slightly above the

unit square (0 # ~F2

1 # 1 and 0 # ~F2

3 # 1) of the hori-

zontal plane [(w3/w

2) ~F

2

2 5 0], while the second lies

farther from the origin. We will denote by I, II, and III

the three volumes between the two leafs. Flow states

lying on either leaf are considered hydraulically critical

in that at least one neutral long wave of the three-layer

system has zero phase speed.

It remains to determine the hydraulic state of flows

that exist in volumes I, II, and III. To this end, consider

some horizontal [constant (w3/w

2) ~F

2

2 ] slices through

the volumes, as shown in Figs. 4a–c. Begin with the

(w3/w2)~F2

2 plane, which is intersected by the critical

surfaces along the lines ~F2

1 5 1 and ~F2

3 5 1, as shown

in Fig. 4a. The projections of volumes I, II, and III are

labeled, and it can be seen that II is divided into two

subregions. At finite but small values of (w3/w

2) ~F

2

2 , spe-

cifically (w3/w

2) ~F

2

2 , (11b)�1, region II is no longer

divided (Fig. 4b). For (w3/w

2) ~F

2

2 . (11b)�1, region I

disappears altogether (Fig. 4c).

Now consider first the plane (w3/w2)~F2

2 5 0 in more

detail. All flows lying therein have inactive middle

layers, so the upper and lower layers are decoupled and

act essentially as single layers.Were the layer depths and

velocities uniform in y, the upper layer would support

two long gravity waves with speeds u1 6 (g921H1)1/2, while

the lower layer would have its own pair of waves with

speeds of u36 (g931H3)1/2. In the unit square region (I) near

the origin ( ~F2

1 5 u21/g912H1 , 1, ~F2

3 5 u23/g923H1 , 1), the

two upper-layer waves propagate in opposite directions,

as do the two lower-layer waves. The flow is therefore

hydraulically subcritical with respect to both upper-

layer and lower-layer modes. It is easy to show that both

waves belonging to the upper layer propagate in the

same direction in region IIa, whereas the two lower-

layer waves propagate in opposite directions. Region IIa

therefore corresponds to supercritical flow with respect

to just the upper-layer mode only. The reverse is true in

region IIb. In region III the flow is supercritical with

respect to both modes, meaning that all four waves

propagate in the same direction, or that the two upper-

layer waves propagate in one direction and the two

lower-layer waves propagate in the other.


It is now clear that volume I of Fig. 3 is bordered

below by the subcritical region of the (w3/w2)~F2

2 plane

just described. It is not difficult to show that this volume

is also bordered on its sides (planes ~F2

1 5 0 and ~F2

3 5 0)

by regions of flow that would be considered subcritical

under traditional conditions (cross-strait independent

velocities and layer thicknesses). We will therefore call

states laying within volume I provisionally subcritical.

The wave modes that arise in the interior of volume I

generally cannot be decoupled into upper-layer and

lower-layer modes as they were above, so it is not pos-

sible to refer to the upper or lower layer as being sub-

critical. Instead, one must speak of subcriticality with

respect to the two wave modes, which are related to the

first and second baroclinic modes of a continuously

stratified flow. Similarly, flow states laying within volume

II will be categorized as provisionally supercritical with

respect to one mode. Again, this volume is bordered by

regions that, under traditional conditions, would be su-

percritical with respect to one of the internal modes, but

not the other. Volume III is considered provisionally

supercritical with respect to both modes. Classification

of the flow in any region is further complicated by the

presence of lateral shear, which may give rise to wave

modes not present in the traditional three-layer system.

Also, imaginary phase speeds (long-wave instability) are

common within both of the supercritical regions, even

in the traditional case. In interpreting the critical con-

ditions (2) or (3), or Figs. 3 and 4, it should be kept in

mind that the long-wave speeds depend on the shear

between and across the layers, and cannot generally be

expressed as the sum of a current speed and a resting

propagation speed. Thus, the Froude numbers ~F1, ~F

2,

and ~F3 are defined for convenience and do not have a

simple meaning in terms of propagation speeds or their

components.

To locate the volume that a particular flow state lies

within, first calculate the values of ~F2

1 , (w3/w2) ~F2

2 , and~F2

3 .

LetZ represent (w3/w

2) ~F

22 , and letZ

c(~F21 ,

~F23 ) represent

the critical surface defined by (5). If

~F2

3 ,11b

b�

~F2

1

b, (6)

then Z , Zc means that the flow is provisionally sub-

critical, whileZ.Zcmeans that the flow is provisionally

supercritical with respect to just one mode; whereas, if

~F2

3 .11b

b�

~F2

1

b, (7)

then Z , Zc means that the flow is provisionally su-

percritical with respect to one mode, while Z . Zc

means that the flow is provisionally supercritical with

respect to both modes (note that Zc may be ,0).

FIG. 3. Critical surface for b 5 0.25.


One result that may be relevant to the Strait of Gi-

braltar is that if the intermediate layer experiences some

interval across the strait over which the velocity u2 is

very small, then (w3/w2)~F2

2 will be quite small and the

upper and lower layers become decoupled. Thus, de-

coupling of the upper and lower layer does not require

that u2 be zero across the entire strait, at least not in

terms of hydraulic control of the flow.

c. Further remarks on the behavior of longwaves and stability

This section attempts to develop further insight into

the meaning of volumes I, II, and III of Fig. 3. An un-

interested reader may skip this discussion and proceed

to section 3. We examine the wave speeds themselves

and, for simplicity, consider the traditional setting in

which the channel cross section is rectangular (w1 5w2 5 w3) and ui are independent of y. Under these

conditions ~Fi reduces to the ordinary Froude number for

each layer. The long-wave speeds can be calculated from

Eq. (2.7) of Pratt et al. (1999) and can be shown to de-

pend on the three Froude numbers as well as the value of

r and the dimensionless depth ratios d15H1/H and d25H2/H, where H 5 H1 1 H2 1 H3.

As a first example, consider the variations of phase

speeds along a line defined by ~F2

1 5 ~F2

2 5 ~F2

3 in the

space of Fig. 3. One can begin at the origin and follow

this line outward as it cuts through the two surfaces

separating volumes I, II, and III. The phase speeds de-

pend on the signs of ~Fi, which we take to be positive, so

the flow is unidirectional. We also assume that r 5 0.5

and that the layer depths are equal (d15 d25 1/3) so that

the layer velocities are equal and there is no interfacial

shear. The results are plotted in Fig. 5a as a function of~F1. The system contains two pairs of internal long waves,

and we denote their nondimensional speeds (scaled by

g923H) c16 and c2

6. At ~F15 0 the background state is

quiescent and each pair of speeds is evenly distributed

around zero. Here c16 are easily identifiable as the speeds

of the first baroclinic mode. Their magnitude exceeds

that of the speeds c26 of the second baroclinic mode. As

~F1 increases, the flow moves as a slab with an increasing

velocity and the modes are advected in the same direc-

tion. The system first becomes supercritical with respect

FIG. 4. Slices of constant (w3/w2)~F2

2 for b 5 0.25.


to the second mode near ~F1 5 0.56, and with respect to

both modes near ~F1 5 1. The vertical lines that mark

these transitions coincide with the boundaries separat-

ing volumes I, II, and III.

An example that is more realistic for exchange flow

has the same settings above but with ~F35 �0.6 ~F

1and

~F25 0.1 ~F

1. The presence of shear between the layers

makes long-wave instability possible and the real and

imaginary parts of the wave speeds are shown separately

in Figs. 5b and 5c. Near ~F1 5 0 the modes are stable and

separated into two pairs. As ~F1 increases, c�2 undergoes

a zero crossing and the subcritical flow becomes super-

critical with respect to the second internal mode. At a

slightly larger value of ~F1, c�2 and c2

1 briefly merge,

producing a small band of long-wave instability. The

positive imaginary part of c2 is plotted in Fig. 5c, and the

instability appears as a small mound of positive values

about ~F1 5 1.25. At slightly larger ~F1 the two modes

separate and the flow once again becomes stable and

supercritical with respect to one mode. However, two

other waves merge near ~F1 5 1.6, and the flow again

becomes unstable. At slightly larger ~F1, one of the

neutral modes undergoes a zero crossing and this marks

the transition from region II to region III. In the latter,

there is one unstable mode pair, each member having

the same real speed (cr) but with equal and opposite

imaginary parts, and one pair of neutral waves with

negative phase speeds. This pair later merges and be-

comes unstable as well. As the reader can see, the journey

from region I to III is complex, with mergers between

different wave components and resulting instabilities. At

the boundaries between the regions, at least one neutrally

stable, stationary long wave exists, but the other waves

may have imaginary speeds.

3. Numerical model description

The numerical model used for this study is the three-

dimensional s-coordinate Princeton OceanModel (POM)

designed in the late 1970s by Blumberg and Mellor

(1987) to study both coastal and open ocean circulation.

The model uses a curvilinear orthogonal grid covering

the region between the Gulf of Cadiz and the Alboran

Sea (Fig. 6). The grid has a nonuniform horizontal

spacing; horizontal resolution is higher in the strait, where

it is around 300 m, with respect to the eastern (western)

ends where it reaches 10–20 km (8–15 km). The portion

of the horizontal grid representing the strait is rotated

anticlockwise ;178 so that the along-strait velocity is

quitewell represented by themodelu component (Fig. 7).

The vertical grid is made of 32 sigma levels, logarithmi-

cally distributed at the surface and the bottom and uni-

formly distributed in the rest of the water column. Model

topography has been obtained by merging the 2-min

FIG. 5. Phase speeds for the four long waves of a traditional

three-layer system with no cross-channel variations in layer

thicknesses or velocities. (a) The phase speeds as functions of ~F1

for the unidirectional slab flow ~F2

1 5 ~F2

2 5 ~F2

3 , d1 5 d2 5 1/3and r 5 0.5. (b) Real and (c) imaginary parts of the waves speeds

are plotted for the same r and dn, but under the exchange flow

conditions ~F3 5 �0.6 ~F1 and~F2 5 0.1 ~F1.

FIG. 6. Horizontal model grid.


Gridded Global Relief Data (ETOPO2) bathymetry

(U.S. Department of Commerce and NOAA/NGDC

2001) with the very high resolution digitalized bathyme-

try provided by the PhysicalOceanographicGroup of the

University of Malaga. To reduce the well-known pres-

sure gradient error produced by sigma coordinates in

regions of steep topography (Haney 1991), a smoothing

was applied in order to reach values of dH/H, 0.2, where

H is themodel depth, as suggested byMellor et al. (1994).

The resultingmodel topography in the region of the strait

is shown in Fig. 8. Two open boundaries are defined at the

eastern and western ends of the computational domain;

here, an Orlanski radiation condition (Orlanski 1976) is

used for the depth-dependent velocity, while a forced

Orlanski radiation condition (Bills and Noye 1987) is

used for the surface elevation and a zero gradient con-

dition for the depth-integrated velocity. Boundary con-

ditions for both temperature and salinity are specified by

using an upwind advection scheme that allows advection

of temperature and salinity into the model domain under

inflow conditions. Normal velocities are set to zero along

coast boundaries, while at the bottom an adiabatic

boundary condition is applied to temperature and sa-

linity; finally, a quadratic bottom friction with a pre-

scribed drag coefficient is applied to themomentum flux.

The Smolarkiewicz upstream-corrected advection scheme

for tracers (Smolarkiewicz 1984), as implemented by

Sannino et al. (2002), was used in the present study. The

model resolves the vertical subgrid-scale turbulence

by prognostic equations for the turbulent velocity and

length scale (Mellor and Yamada 1982); thus, there is

no need for specific parameterizations of entrainment,

as recently demonstrated by Ezer (2005). This feature

makes our model capable of taking into account the

effect of entrainment and mixing between Atlantic and

Mediterranean waters. The model starts from rest and is

forced at the open boundaries through the specification

of the surface tidal elevation characterized by the prin-

cipal two semidiurnal and two diurnal harmonics: M2,

S2,O1, andK1. Amplitude and phase of these harmonics

have been computed via the OTIS package (Egbert and

Erofeeva 2002). Finally, the initial conditions in terms of

salinity and temperature have been taken from the Med-

iterranean Data Archaeology and Rescue (MEDAR)/

Mediterranean Hydrological Atlas (MEDATLASII)

climatologic Mediterranean and Black Sea Database

(MEDARGroup 2002) for the month of April. The pres-

ent model can be considered as an improved version of

the model implemented by Sannino et al. (2007) as it is

characterized by a better resolved bathymetry and more

realistic initial and boundary conditions.

The model was initially run for 240 days without tidal

forcing in order to achieve a steady two-way exchange

system. Then, the model simulation was extended for

another 7 days forced by tidal components in order to

achieve a stable time periodic solution. Finally, the

model was run for a further tropical month (27.321 days)

that represents our main experiment. The term ‘‘time

averaged’’ that will be used in the following refers to

the average over this tropical month period. Spring tide

corresponds to day 21 of the simulation; neap tide to

day 27.

Model validation

A complete validation analysis of the numerical

model has been performed by Sanchez-Roman et al.

(2009). In their work they compared the predicted and

observed amplitude and phase of the diurnal and semi-

diurnal tidal components of the along-strait velocity

field; in particular, they first collected data for the month

ofApril from different observed datasets along the strait

and from the model simulation for the same locations,

and then applied to them the classical Foreman vectorial

harmonic analysis (Foreman 1978; Pawlowicz et al.

2002). The results obtained were considered satisfactory,

FIG. 7. Horizontal model grid for the region of the strait.

FIG. 8. Model bathymetry for the region of the strait. Gray levels

indicate the water depth. The point ES and CS represent the po-

sition for Espartel Sill and Camarinal Sill, respectively. White lines

indicate sections used to present model results.


with differences limited in most of the strait to less than

10 cm s21 in amplitude and 208 in phase.

Good agreement between observed and model data

also has been found for the surface elevation. In Tables

1 and 2 the observed amplitudes (A) and phases (P) of

the two semidiurnal tidal components [that represent

more than 80% of the total tidal signal according to

Candela et al. (1990)] of the surface elevation are

compared with the simulated amplitudes (A) and phases

(P) of the same components. Here one can see that the

maximumdifferences do not exceed 3.6 cm in amplitude

(with a maximum error that however does not exceed

18%) and 118 in phase.

The model is also able to reproduce the generation

and subsequent propagation of the internal bores with

characteristics similar to those described by FA88 (see

Fig. 9). About 1 h before high tide at Tarifa the internal

bore is released from CS (Fig. 9a) and starts to travel

eastward. The bore is released when the upper layer

starts to move eastward, while the lower layer continues

TABLE 1. Comparison between observed and predicted amplitudes A and phases P of the M2 tidal elevation.

Observed M2 Predicted M2 Predicted 2 observed

Location Lat Lon A (cm) P (8) A (cm) P (8) A (cm) A (%) P (8)

Tsimplis et al. (1995)

Gibraltar 368089 058219 29.8 46.0 29.5 46.0 20.3 1.0 10.0*

Garcıa-Lafunete (1986)

Punta Gracia 36805.49 05848.69 64.9 6 0.2 49.0 6 0.5 67.6 53.8 12.7 4.1 14.5

Tarifa 36800.29 05836.49 41.5 6 0.2 57.0 6 0.5 43.5 49.7 12.0 4.8 27.3

Punta Cires 35854.79 05828.89 36.4 6 0.2 46.5 6 0.5 35.0 54.9 21.4 3.8 18.4

Punta Carnero 36804.39 05825.79 31.1 6 0.2 47.5 6 0.5 30.8 47.4 20.3 0.9 20.1

Candela et al. (1990)

DN 358589 058469 60.1 51.8 58.2 57.8 21.9 3.1 16.0

DS 358549 058449 54.0 61.8 54.1 64.1 10.1 0.2 12.3

SN 368039 058439 52.3 47.6 52.3 52.9 0.0 0.0 15.3

SS 358509 058439 57.1 66.8 56.8 67.4 20.3 0.5 10.6

DW 358539 058589 78.5 56.1 76.6 62.7 21.9 2.4 16.6

TA 368019 058369 41.2 41.2 43.5 49.7 12.3 5.5 18.5

AL 368089 058269 31.0 48.0 30.0 49.7 21.0 3.2 11.7

CE 358539 058189 29.7 50.3 29.5 51.5 20.2 0.6 11.2

DP5 368009 058349 44.4 47.6 42.1 47.6 22.3 5.1 10.0

* Calibration.

TABLE 2. Comparison between observed and predicted amplitudes A and phases P of the S2 tidal elevation.

Observed S2 Predicted S2 Predicted 2 observed

Location Lat Lon A (cm) P (8) A (cm) P (8) A (cm) A (%) P (8)

Tsimplis et al. (1995)

Gibraltar 368089 058219 10.7 72.0 11.6 72.0 10.9 8.4 10.0*

Garcıa-Lafuente (1986)

Punta Gracia 36805.49 05848.69 22.3 6 0.2 74.0 6 1.0 25.9 77.6 13.6 16.1 13.6

Tarifa 36800.29 05836.49 14.2 6 0.2 85.0 6 1.5 16.8 73.9 12.6 18.3 211.1

Punta Cires 35854.79 05828.89 14.1 6 0.2 74.0 6 1. 14.5 81.2 10.4 2.8 17.2

Punta Carnero 36804.39 05825.79 11.5 6 0.2 71.0 6 1.0 12.1 72.0 10.6 5.2 11.0

Candela et al. (1990)

DN 358589 058469 22.5 73.8 22.4 81.4 20.1 0.4 17.6

DS 358549 058449 21.1 83.3 20.7 88.2 20.4 1.9 14.9

SN 368039 058439 18.5 73.4 20.6 76.3 12.1 11.3 12.9

SS 358509 058439 20.6 92.3 21.9 91.0 11.3 6.3 21.3

DW 358539 058589 29.0 82.2 29.2 85.4 10.2 0.7 13.2

TA 368019 058369 14.7 67.9 17.3 72.8 12.6 17.7 14.9

AL 368089 058269 11.1 73.9 11.7 75.0 10.6 5.4 11.1

CE 358539 058189 11.4 75.6 11.8 78.1 10.4 3.5 12.5

DP5 368009 058349 16.1 73.9 16.2 72.6 10.1 0.6 21.3

* Calibration.


to move westward. Its initial length scale, in the along-

strait direction, is about 3 km and its travel times from

CS to Tarifa, Punta Cires, andGibraltar sections are 2, 4,

and 6 h, corresponding to a speed of about 1.7 m s21

between Camarinal Sill and Tarifa, 2.5 m s21 between

Tarifa and Punta Cires, and 1.5 m s21 between Punta

Cires and Gibraltar sections. The amplitude of the

eastward propagating bore decreases progressively from

about 100 m on the western edge of CS to about 50 m at

the Gibraltar section. Initially the bore is characterized

by two large and steep internal waves that during the

eastward propagation disintegrate, at the exit of the

strait, into dispersive wave trains.

However, what is observed in the Strait of Gibraltar is

that the bore, at the eastern exit of the strait, disinte-

grates into rank-ordered sequences of internal solitary

waves followed by a dispersive wave (Brandt et al.

1996). The model is not able to reproduce these internal

solitary waves as nonhydrostatic effects are neglected.

However, given that most of the principal bore charac-

teristics are well simulated by the model both at CS and

along TN, it can be considered as a suitable tool for

FIG. 9. (a)–(f) Evolution of salinity pertubations during spring tide. Contours are shown with an interval of 0.5 psu. The snapshots are

plotted at an interval of 1 h. The time moments are referred to the surface elevation at Tarifa.


studying the criticality of the flow exchange in the Strait

of Gibraltar.

4. Results

In this section output obtained from running the main

numerical experiment will be analyzed on the basis of

the hydraulic condition described in section 2. The

analysis will be conducted first for the two-layer and

subsequently for the three-layer case. Note that the

cross-strait layer velocity un and the layer density rnused in the following hydraulic analysis are computed as

un(x, y, t)5

ðupn(x,y,t)

dwn(x,y,t)

u(x, y, z, t) dz, (8)

rn(x, t)5

ðyNn(x)

ySn(x)

ðupn(x,y,t)dw

n(x,y,t)

r(x, y, z, t) dz dy, (9)

where upn and dwn are the instantaneous depths of the

upper and lower bounds of the nth layer and ySn and yNn

represent the southern and northern limit of the cross

section x and nth layer.

Owing to the bathymetric structure of the western

part of the strait, the hydraulic analysis in that region

will be restricted to the southern channel only.

a. Two-layer approximation

To identify regions where the flow is hydraulically

controlled in the two-layer approximation, the instan-

taneous generalized composite Froude number (Gw2 ) is

evaluated following Eq. (2) and considering r1 and r2 as

the mean density of the upper and lower layer, respec-

tively. To carry out the analysis it is necessary first to

define an interface betweenMediterranean andAtlantic

waters. The method used consists in defining the inter-

face as the tropical month-averaged salinity surface

corresponding to the zero tropical month-averaged ve-

locity field. The internal salinity surface obtained is

shown in Fig. 10.As expected, the salinity increases from

west to east changing from about 37.25 psu at CS up to

about 38.1 psu at the east entrance of the strait. The

value 37.25 psu is in agreement with those used by

Bryden et al. (1994) and Candela et al. (1989) to com-

pute the volume transport across a section passing

through CS, while the value 38.1 psu is the same as

adopted by Candela et al. (1989) and Baschek et al.

(2001) to compute the transport through a cross section

at the eastern end of the strait. Using this material in-

terface the instantaneous generalized composite Froude

number (Gw2 ) has been computed. Looking at Fig. 11a,

which shows the frequency of occurrence of supercritical

composite Froude number over the tropical month pe-

riod, it appears evident that the frequency is low in TN,

while higher values are reached in the region of CS

and ES; in particular, at CS the frequency is 50% while

west of ES it is about 20%. As expected the critical con-

dition in TN is primarily due to the upper-layer flow

( ~F2

1 � ~F2

2 ), while at ES and CS it is primarily due to the

lower-layer flow.

While the previous description is useful in identifying

the regions within the strait where the flow becomes

supercritical, a complete understanding of the hydraulic

regimes, in terms of maximal or submaximal exchange,

can be achieved only by exploring the simultaneous

presence of supercritical flow regions through the strait.

This is done in Fig. 12, where, in the light of the previous

findings, the analysis is restricted over the two sills of CS

and ES and the region of TN. The bars indicate the

presence of supercritical flow in those regions. It can be

shown that the lower-layer Froude number is dominant

at ES and CS, the upper layer being relatively inactive,

while the upper layer is dominant at TN. At CS the flow

is supercritical two times per semidiurnal period except

during neap tide when it becomes supercritical only

once per diurnal period. In general, the flow reaches a

supercritical state during high water in Tarifa, and then

the control is lost and recovered again during the sub-

sequent rising tidal phase. Criticality of the flow at ES

shows a quite different behavior during spring and neap

tide. During spring tide (from day 7 to 10 and from day

16 to 23) the flow appears to be supercritical for every

FIG. 10. Time-averaged interfacial-layer salinity for the two-layer

approximation.


descending tidal phase at Tarifa, and then the control is

lost during low water in Tarifa and recovered again

during the subsequent rising tidal phase. Due to the di-

urnal inequality, the flow is supercritical during neap

tide only one time per day, while the supercritical con-

dition disappears completely during neap days 27 and

28. A similar general behavior has been found also along

TN: the flow is supercritical during the descending tidal

phase at Tarifa for every semidiurnal tidal period

around spring days, while the duration of supercritical

conditions is reduced for every diurnal tidal period

during neap days, disappearing completely during neap

days from 26 to 31.

What emerges from the previous description is the ab-

sence of any permanent supercritical region along the

strait; however, this seems to be in contrast with the sim-

ulated character of the flow. For example, in the region of

ES the Mediterranean water is always directed westward

and subject to a permanent hydraulic jump west of ES.

This is confirmed by the abrupt deepening of the isoha-

lines west of ES and by the presence of mixing, which is

evidenced by a Richardson number, defined as

FIG. 11. Frequency of occurrence of supercritical flow for the (a) two-layer and (b) three-layer approximation.

FIG. 12. Bars indicating the presence of supercritical condition, for the two-layer case, in the three main regions of

the Strait: ES, CS, and TN. For ES and CS the critical condition is referred to the lower layer, while for TN it is

referred to the upper layer. (bottom) Tidal elevation at Tarifa.


Ri5� g

r0

›r

›z

du

dz

� ��2

, (10)

which is always less than the critical value of 0.25 west of

ES (Fig. 13). Moreover, looking at the salinity distribu-

tion along cross sectionD, one can note that theAtlantic

flow is cyclically detached from the north shore east of

TN, even when the two-layer hydraulic model predicts a

submaximal flow inTN (Fig. 14). As initially suggested by

FA88, and subsequently demonstrated both by Bormans

et al. (1986) and Timmermans and Pratt (2005), the

separation of the flow from the north shore east of TN is

the clear indication that the Atlantic flow has reached a

supercritical state. Both evidences reinforce the hypoth-

esis initially put forward by Sannino et al. (2002, 2007)

that the direct application of the two-layer hydraulic

theory to the Strait of Gibraltar is not obvious.

b. Three-layer definition and properties

For our three-layer framework we follow BR95 and

use the upper and lower limit of the halocline as inter-

faces. However, a different quantitative method for di-

viding all salinity profiles into Atlantic layer (AL),

interfacial layer (IL), and Mediterranean layer (ML) is

used. Salinity profiles are fitted with a hyperbolic tan-

gent; in particular, the upper and lower bounds of the

interfacial layer are represented by the intersections of

the tangent at the inflection point of the hyperbolic tan-

gent with two vertical lines passing respectively through

the simulated salinity at the surface and at the deepest

FIG. 13. (a)–(f) Velocity fields along section E during neap tide. Triangles (both black and white) mark the position where the flow

reaches a Richardson number less than 0.25. (bottom) Times of the individual snapshots marked on the tidal elevation at Tarifa. Contour

lines indicate the position of the isohaline.


salinity (defined as the arithmetic mean between the

deepest simulated salinity and the deepest fitted salinity).

To quantitatively measure the fit quality of all salinity

profiles, the amount of variance that is explained by the

hyperbolic tangent fitting was computed as

fq[ 1� �k(S

k� S

k)2 �

k(S

k� S)2

� �� 3 100,

�(11)

where k represents the model vertical levels, Sk is the

simulated salinity, Skis the fitted salinity, and S is the

arithmetic mean of the profiles. Only salinity profiles

with a fit quality .98% are retained; they however

represent more than 90% of the available salinity pro-

files (about 3500). The mean fit quality value obtained

averaging over the entire set of retained salinity profiles

is about 99.5%. As an example, three different salinity

profiles and the respective fitted curves are shown in

Fig. 15.

Figure 16 shows the time-averaged thicknesses of

the three layers together with the depth of the midpoint

of the interfacial layer. The thickness of AL reduces

gradually from west to east except over CS where it

undergoes a more evident reduction, halving its value.

At the eastern end of TN the effect of rotation becomes

FIG. 14. (a)–(f) Salinity fields along section D during neap tide.


important and the AL shows a strong cross-strait re-

duction from the southern to the northern shore.Amore

dramatic reduction is suffered by ML, which decreases

from about 200 to about 50 m over CS. This strong re-

duction is ascribed to the strongmixing generated by the

hydraulic jump that takes place every semidiurnal tidal

period west of CS. As pure MediterraneanWater passes

over CS, it flows principally along the deep southern

channel where it undergoes another mixing due to the

hydraulic jump present west of ES. In confirmation of

the strong mixing between the two water masses of

Atlantic and Mediterranean origin, IL thickness can

reach more than 140 m with peaks of about 180 m in the

regions where hydraulic jumps are present. The com-

puted midpoint depth of the interfacial layer is in very

good agreement with that shown by BR95 (see their

Fig. 6); however, the interfacial layer thickness in our case

is systematically higher. This difference can be principally

attributed to the fact that the analysis of BR95 does not

take explicitly into account the tidal variability, while we

use simulated data from an entire tropical month tidal

period. Another reason is that BR95 analyzed salinity

profiles obtained by averaging data collected in No-

vember 1985, March 1986, June 1986, and October 1986,

while our simulation is performed on a climatological

month of April.

Other interesting features can be observed looking at

the temporal variability of the interfacial layer thickness

that is strongly modulated by tide. Figure 17 shows the

interfacial layer thickness as a function of longitude and

time. It is referred to section E and covers an entire day

during spring and neap tide. Here the presence of the

two hydraulic controls west of CS and ES is evident. The

interfacial thickness west of ES does not show any sig-

nificant variability, both in space and time, during spring

and neap periods. This is, again, a clear indication that

the hydraulic jump west of ES is a permanent feature.

On the other hand, the interfacial thickness in the region

of CS shows a periodic shift of the hydraulic jump from

west to east, and vice versa at diurnal frequency.

Moreover, the amplitude of the thickness is strongly

modulated during the tropical month; in particular,

during spring tide it reaches about 200 m on both sides

of CS, while during neap tide this value is reached just on

the west side, very close to the sill. Finally, the thickness

of the interfacial layer at CS shows variability owing to

the diurnal inequality of the tide; this is more evident

during neap tide.

The instantaneous Atlantic layer transport (ALT), in-

terfacial layer transport (ILT), and Mediterranean layer

transport (MLT) for each model cross section within the

strait have been computed according to

ALT(x, t)5

ðyN1(x)

yS1(x)

ðup1(x,y,t)dw1(x,y,t)

u(x, y, z, t) dz dy, (12)

ILT(x, t)5

ðyN2(x)

yS2(x)


u(x, y, z, t) dz dy, (13)

MLT(x, t)5

ðyN3(x)

yS3(x)


u(x, y, z, t) dz dy. (14)

The resulting transports over the tropical month period

are shown in Fig. 18 for four different cross-strait sec-

tions located at ES, CS, Tarifa, and Gibraltar, respec-

tively (sections A, B, C, and D in Fig. 8). The Atlantic

layer carries water principally toward the east with a

FIG. 15. Comparison between the simulated (solid lines) and the fitted (dashed lines) salinity profiles. The tangent (dashed

dotted line) to the flex of the fitted profile and the resulting interfacial layer (gray region) are also plotted.


small fraction of the transport cyclically directed also in

the opposite direction. This small fraction decreases

progressively from west to east becoming null after

crossing CS. An opposite behavior is exhibited by MLT

where the principal direction is toward the west, and the

eastward fraction decreases gradually from section D to

section A where it is reduced to zero. Figure 18 also

shows that ILT reaches values comparable both with

ALT andMLT, contributing to transport-mixedAtlantic–

Mediterranean water alternatively in both directions ev-

erywhere along the strait. It is noteworthy that the ILT

amplitude exceeds the ALT and MLT amplitude at the

eastern and western end of the strait, respectively.

c. Three-layer hydraulics

For the three-layer case it is interesting first to ask

whether the interfacial layer experiences some interval

across the strait over which (w3/w2)~F2

2 is so small that

the upper and lower layers are decoupled. Looking for

time intervals in which (w3/w2)~F2

2 # 10�4 over the en-

tire tropical month period, it appears that the upper and

lower layers are decoupled only for less than 10% of

time everywhere in the strait with minimum values

limited to the west of ES, around CS and in the eastern

part of TN. Moreover, during these time intervals both

upper and lower layer flows are always subcritical. Thus,

in order to identify regions where the flow is provision-

ally supercritical with respect to one or both modes,

the instantaneous values of ~F2

1 , (w3/w2)~F2

2 , and~F2

3 are

computed following (4), and then the rules defined by

(6) and (7) are applied. (For simplicity, the terms su-

percritical and subcritical will hereafter be used instead

of provisional supercritical and provisional subcritical.)

Results are summarized in Fig. 11b, where the frequency

FIG. 16. Time-averaged Atlantic, interfacial, and Mediterranean layer thickness, and depth of the midpoint of the

interfacial layer as simulated by the numerical model.


of occurrence over the tropical month period of super-

critical flow with respect to at least one mode is shown.

Here one can note that supercriticality is reached with

higher frequency relative to the two-layer case at ES and

TN. In particular, at the eastern end of TN the frequency

reaches more than 60%, while west of ES, in accordance

with the previous consideration on the flow character-

istics, the frequency is about 100%.

Further insight on hydraulics can be gained by looking

at the temporal variability of the controlled flow re-

stricted, in the light of the previous findings, to the

region of TN and around the two sills of ES and CS

(Fig. 19). Black bars in Fig. 19 indicate the presence of

supercritical flow due to just one mode, while gray bars

mark the instants when the flow is supercritical with

respect to bothmodes. Except aroundESwhere the flow

is permanently supercritical, the flow is only intermit-

tently supercritical both in CS and along TN during each

diurnal tidal cycle. The frequency of appearance of su-

percritical flow, with respect to just one mode and with

respect to both modes, over the entire tropical month

period is about 76% and 67% at TN and CS, respec-

tively. Moreover, while the flow is supercritical with

respect to bothmodes for only 6% in TN, the percentage

increases up to 26% at CS. A similar percentage is also

found at ES. Thus, while at CS the flow is supercritical

with respect to one and both modes with approximately

the same percentage, the flow is principally controlled

with respect to only one mode both at ES and TN.

The control along TN always starts to develop at its

eastern boundary when the tide is high at Tarifa. During

the subsequent descending phase the control extends

more and more toward the west along TN, reaching

maximum extension after about 4 h. In the remaining

descending phase, and also during the subsequent rising

phase, that is, when the magnitude of the velocity in the

Atlantic layer decreases, the control is progressively lost

along TN starting from its western boundary. During

spring periods, that is, from day 3 to 8 and from day 16 to

22, the control is completely lost 3 h after low tide, while

for the remaining days the control holds until high tide is

reached in Tarifa. This different behavior is related to

the minimum velocity reached in the upper layer during

each semidiurnal tidal cycle. It is known that the upper

layer, in the region of TN, is always directed toward the

MediterraneanSea.This isdue to theweaknessof the tidal

amplitude relative to the strong mean current at Tarifa.

However, the tide is still able to reduce the magnitude of

the upper-layer flow enough to establish subcritical con-

ditions. This reduction is more evident during spring tide

than during neap tide and, thus, explains why the control

is lost during spring tide and not during neap tide.

At CS the model reproduces the tidally induced pe-

riodic loss and subsequent renewal of the control that

FIG. 17. Interface thickness as a function of longitude and time for the along-strait section E during an entire day

corresponding to (a) spring tide and (b) neap tide. The locations of Camarinal Sill and Espartel Sill are marked by a

black triangle.


occurs two times for each semidiurnal cycle for the en-

tire period except during neap tide (from day 26 to 29).

During rising water at Tarifa the control at Camarinal

starts to develop while, during the subsequent descending

phase, the control is initially lost, then recovered and lost

again during low tide.

From the above description it appears that the ex-

change switches cyclically between a maximal and sub-

maximal regime, with a permanent control at ES and an

intermittent control along TN. In such a configuration

CS is not the principal control of the flow at the western

end of the strait.

5. Summary and concluding remarks

We have investigated the hydraulic behavior of the

exchange flow through the Strait of Gibraltar using the

recent hydraulic criterion developed by Pratt (2008).

FIG. 18. Time-dependent Atlantic layer (blue line), interfacial layer (green line), and

Mediterranean layer (red line) volume transports at (a) Espartel Sill, (b) Camarinal Sill, (c)

Tarifa (TA), and (d) Gibraltar (GI), respectively, Figs. 8a–d.


The continuous vertical stratification has been approx-

imated first by a two-layer system and then by a three-

layer system and cross-strait variations in velocity and

layer thickness are accounted for. In the two-layer case

the interface has been chosen approximately in the

middle of the layer in which entrainment and mixing

occur between Mediterranean and Atlantic waters,

while for the three-layer case this transition zone has

represented one of the three layers. We have focused

our attention on the differences induced by the two

different vertical approximations on the predicted hy-

draulic regime in the strait. A three-dimensional nu-

merical simulation reproducing the two-way exchange

in the strait at high resolution and tidal frequency has

provided data for the hydraulic analysis; in particular,

the simulation covered an entire tropical month period.

The hydraulic behavior analyzed in the two-layer

framework has predicted intermittent controls at CS, ES,

andTN, with a frequency of occurrence of 50%, 23%, and

15% respectively, lower than indicated by previous stud-

ies. Comparisons must bemade with caution, for previous

studies are based on the local composite Froude number,

often measured near the channel centerline. In reality,G2

can vary strongly across the channel (Fig. 20), sometimes

ranging above and below unity across the same section. Its

value at any particular location is not an indicator of hy-

draulic control. In the TN G2 remains ,1 in the channel

center at all times, occasionally exceeding unity along

the sides where the shoaling depth causes the layer

thickness to decrease. A simple inspection of Fig. 20

does not determine whether the flow at TN is hydrauli-

cally subcritical or supercritical; only evaluation of the

cross-strait integral condition (2) can decide that question.

Comparisons with previous work must also take into

consideration differing definitions of the two-layer in-

terface. For example, FA88 found a permanent hydraulic

control at TN, a conclusion that is in disagreement with

Fig. 20 showing values of G2 permanently ,1 in the

channel center at TN. As an interface at the eastern end

of the strait, FA88 used the surface su 5 28.0, which

corresponds approximately to a salinity of 37.4 psu, and

as velocities the values at middepth of each layer. The

choice of su 5 28.0 is problematic for the eastern part of

the strait because it determines an upper layer that is too

thin, giving rise to a general overestimation of the upper-

layer velocity and overestimation of G2.

FIG. 19. Bars indicating the presence of provisional supercritical flow with respect to one mode (black) and with

respect to both modes (gray) in the three main regions of the strait: Espartel Sill, Camarinal Sill, and Tarifa Narrow.

(bottom) Tidal elevation at Tarifa.

FIG. 20. Maps of the frequency of occurrence of supercritical

local composite Froude numbers for the two-layer one-dimensional

case.


That the choice of the interface between the two layers

may alter the values of G2 was already argued by Send

and Baschek (2001). They computed the Froude number

at the eastern end of the strait, using both the 37.4 and

the 38.1 isohaline, as the interface during a completeM2

tidal cycle for both spring and neap tide during April

1996. For both computations the velocities of the two

layers were determined by taking the mean value away

from the main shear zone. The results showed that using

the 37.4 isohaline as the interface led the flow to appear

controlled during part of the tidal cycle, both during neap

and spring tide, while using the 38.1 isohaline caused

the flow to always appear subcritical. The ambiguity in

the computation of the Froude numbers is therefore due

in part to the difficulty of fitting a two-layer model to

a flow with a substantial interfacial layer. In some cases,

investigators are content to estimate the Froude number

of the upper (or lower) layer above. This approach cor-

responds to the implicit assumption that the transition

layer is stagnant, implying that theAtlantic layer and the

Mediterranean layer are decoupled. However, this is not

the case in the Strait of Gibraltar where, as we have

demonstrated in section 2, the upper and lower layer are

decoupled for only 10% of the total tropical month pe-

riod and, moreover, during these periods neither of the

layers appear to be supercritical at TN.

The analysis carried out by means of the three-layer,

two-dimensional hydraulic model has shown substantial

differenceswith respect to the two-layer, two-dimensional

model. In the three-layer case the frequency of occur-

rence of the intermittent controls, both at CS and TN, is

increased up to 67% and 76%, respectively, but, more

importantly, on the west side of ES the three-layer model

predicts a permanent supercritical flow. The character of

the simulated exchange flow supports this hydraulic be-

havior: west of ES the model indicates the presence of a

permanent hydraulic jump for the Mediterranean Out-

flow; along the northern shore of TN there is an inter-

mittent detachment of the upper layer.

The hydraulic analysis also indicates that the ex-

change flow in the strait switches cyclically between a

maximal and submaximal regime during the tropical

month period. The maximal regime is achieved when

the flow reaches a supercritical condition along TN since

the flow at the western end of the strait is nearly always

supercritical west of ES. Thus, the simulated flow resulted

in a maximal regime two times per day during spring

tides, and one time per day during neap tide, for a total of

about 76%of the entire tropical month period. It appears

that CS plays only a secondary role with respect to ES.

In summary, there are a number of discrepancies be-

tween our results and previous results for a two-layer

idealization of the Strait of Gibraltar stratification. Our

results generally suggest that control acts less often, and

at fewer locations. As our numerical simulations appear

to match available observations, the discrepancies with

previous work are thought to be due to the previous

neglect of cross-channel variations in the velocity and

layer depth and, more importantly, the ambiguities in

fitting a two-layer model to a system with a substantial

intermediate layer. Moreover, when we fit our contin-

uously stratified model results to a three-layer system

with an active intermediate layer, the results of the hy-

draulic analysis become more consistent with conven-

tional wisdom about the exchange flow (i.e., that it is

predominantly maximal) and with the qualitative be-

havior of the model itself.

For a more complete understanding of the hydraulic

criticality of the exchange flow in the Strait of Gibraltar,

one must also account for the meteorological subinertial

forcing together with the seasonal long-term forcing.

Moreover, the interpretation of maximal and submaxi-

mal hydraulic control has been discussed by past authors

mainly in the context of two-layer systems that allow no

mass exchange between the layers. The Strait of Gi-

braltar situation, with a third layer and with mass ex-

change between layers, is more problematic. In labeling

the exchange ‘‘maximal’’ we have applied the two-layer

criterion that two regions of supercritical flow, both

carrying information outward (away from the strait),

exist. In a three-layer system, one of those regions may

be supercritical with respect to a first baroclinic mode

and the other with respect to a second baroclinic mode.

This situation raises questions of interpretation that

would require a formal hydraulic theory for three-layer

model that allows for mass exchange between layers.

Such a model would be quite ambitious and is beyond

the scope of the present work.

Acknowledgments. We thank Jesus Garcıa-Lafuente,

Jose Carlos Sanchez-Garrido, Antonio Sanchez-Roman,

and Javier Delgado for their thoughtful comments

throughout the course of this research, and also pro-

viding us with the very useful digitalized bathymetry of

the strait. The authors would also like to thank the

anonymous referees whose comments helped improve

this manuscript.

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